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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 612491, 17 pages
doi:10.1155/2009/612491
Research Article
Construction of Fixed Po ints by
Some Iterative Schemes
A. El-Sayed Ahmed
1, 2
and A. Kamal
3
1
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
2
Mathematics Department, Faculty of Science, Taif University,
P.O. Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia
3
Mathematics Department, The High Institute of Computer Science, Al-Kawser City, 82524 Sohag, Egypt
Correspondence should be addressed to A. El-Sayed Ahmed,
Received 23 October 2008; Revised 5 February 2009; Accepted 23 February 2009
Recommended by Massimo Furi
We obtain strong convergence theorems of two modifications of Mann iteration processes with
errors in the doubly sequence setting. Furthermore, we establish some weakly convergence
theorems for doubly sequence Mann’s iteration scheme with errors in a uniformly convex Banach
space by a Frech
´
et differentiable norm.
Copyright q 2009 A. El-Sayed Ahmed and A. Kamal. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let X be a real Banach space and let C be a nonempty closed convex subset of X.Aself-
mapping T : C → C is said to be nonexpansive if Tx−Ty≤x − y, for all x,y ∈ C. Apoint
x ∈ C is a fi xed point of T provided Tx x. Denote by FixT the set of fixed points of T;that
is, FixT{x ∈ C : Tx x}. It is assumed throughout this paper that T is a nonexpansive
mapping such that FixT
/
∅. Construction of fixed points of nonexpansive mappings is an
important subject in the theory of nonexpansive mappings and its applications in a number
of applied areas, in particular, in image recovery and signal processing see 1–3. One way
to overcome this difficulty is to use Mann’s iteration method that produces a sequence {x
n
}
via the recursive sequence manner:
x
n1
α
n
x
n
1 − α
n
Tx
n
,n≥ 0. 1.1
Reich 4 proved that if X is a uniformly convex Banach space with a Frech
´
et differentiable
norm and if {α
n
} is chosen such that
∞
n1
α
n
1 − α
n
∞, then the sequence {x
n
} defined
2 Fixed Point Theory and Applications
by 1.1 converges weakly to a fixed point of T. However, this scheme has only weak
convergence even in a Hilbert space see 5. Some attempts to modify Mann’s iteration
method 1.1 so that strong convergence is guaranteed have recently been made.
The following modification of Mann’s iteration method 1.1 in a Hilbert space H is
given by Nakajo and Takahashi 6:
x
0
x ∈ C,
y
n
α
n
x
n
1 − α
n
Tx
n
,
C
n
z ∈ C :
y
n
− z
≤
x
n
− z
,
Q
n
z ∈ C :
x
n
− z, x
0
− x
n
≥ 0
,
x
n1
Pc
n
Q
n
x
0
,
1.2
where P
K
denotes the metric projection from H onto a closed convex subset K of H. They
proved that if the sequence {α
n
} is bounded from one, then {x
n
} defined by 1.2 converges
strongly to P
FixT
x
0
. Their argument does not work outside the Hilbert space setting. Also,
at each iteration step, an additional projection is needed to calculate.
Let C be a closed convex subset of a Banach space and T : C → C is a nonexpansive
mapping such that FixT
/
∅. Define {x
n
} in the following way:
x
0
x ∈ X,
y
n
α
n
x
n
1 − α
n
Tx
n
,
x
n1
β
n
u
1 − β
n
y
n
,
1.3
where u ∈ C is an arbitrary but fixed element in C,and{α
n
} and {β
n
} are two sequences
in 0, 1. It is proved, under certain appropriate assumptions on the sequences {α
n
} and {β
n
},
that {x
n
} defined by 1.3 converges to a fixed point of T see 7.
The second modification of Mann’s iteration method 1.1 is an adaption to 1.3 for
findingazeroofanm-accretive operator A, for which we assume that the zero set A
−1
0
/
∅.
The iteration process {x
n
} is given by
x
0
x ∈ C,
y
n
J
r
n
x
n
,
x
n1
β
n
u
1 − β
n
y
n
,
1.4
where for each r>0,J
r
I rA
−1
is the resolvent of A.In7, it is proved, in a uniformly
smooth Banach space and under certain appropriate assumptions on the sequences {α
n
} and
{r
n
},that{x
n
} defined by 1.4 converges strongly to a zero of A.
Fixed Point Theory and Applications 3
2. Preliminaries
Let X be a real Banach space. Recall that the normalized duality map J from X into X
∗
, the
dual space of X, is given by
Jx
x
∗
∈ X
∗
:
x, x
∗
x
2
x
∗
2
,x∈ X. 2.1
Now, we define Opial’s condition in the sense of doubly sequence.
Definition 2.1. A Banach space X is said to satisfy Opial’s condition if for any sequence {x
k,n
}
in X, x
k,n
ximplies that
lim
k,n→∞
sup
x
k,n
− x
< lim
k,n→∞
sup
x
k,n
− y
∀y ∈ X with y
/
x, 2.2
where x
k,n
xdenotes that {x
k,n
} converges weakly to x.
We are going to work in uniformly smooth Banach spaces that can be characterized by
duality mappings as follows see 8 for more details.
Lemma 2.2 see 8. A Banach space X is uniformly smooth if and only if the duality map J is
single-valued and norm-to-norm uniformly continuous on bounded sets of X.
Lemma 2.3 see 8. In a Banach space X, there holds the inequality
x y
2
≤x
2
2y, jx y,x,y∈ X, 2.3
where jx y ∈ Jx y.
If C and D are nonempty subsets of a Banach space X such that C is a nonempty closed
convex subset and D ⊂ C, then the map Q : C → D is called a retraction from C onto D
provided Qxx for all x ∈ D. A retraction Q : C → D is sunny 1, 4 provided Qx tx −
Qx Qx for all x ∈ C and t ≥ 0 whenever
x tx − Qx ∈ C. A sunny nonexpansive
retraction is a sunny retraction, which is also nonexpansive. A sunny nonexpansive retraction
plays an important role in our argument.
If X is a smooth Banach space, then Q : C → D is a sunny nonexpansive retraction if
and only if there holds the inequality
x − Qx, Jy − Qx≤0 ∀x ∈ C, y ∈ D. 2.4
Lemma 2.4 see 9. Let X be a uniformly smooth Banach space and let T : C → C be a
nonexpansive mapping with a fixed point. For each fixed u ∈ C and every t ∈ 0, 1, the unique
fixed point x
t
∈ C of the contraction C x → tu 1 − tTx converges strongly as t → 0 to a fixed
point of T. Define Q : C → FixT by Qu s−lim
t → 0
x
t
. Then, Q is the unique sunny nonexpansive
retract from C onto FixT; that is, Q satisfies the property
u − Qu, Jz − Qu≤0, ∀u ∈ C, z ∈ FixT. 2.5
4 Fixed Point Theory and Applications
Lemma 2.5 see 10, 11. Let {a
n
}
∞
n0
be a sequence of nonnegative real numbers satisfying the
property
a
n1
≤
1 − γ
n
a
n
γ
n
σ
n
,n≥ 0, 2.6
where {γ
n
}
∞
n0
⊂ 0, 1 and {σ
n
}
∞
n0
are such that
i lim
n →∞
γ
n
0, and
∞
n0
γ
n
∞,
ii either lim
n →∞
sup σ
n
≤ 0 or
∞
n0
|γ
n
σ
n
| < ∞.
Then, {a
n
}
∞
n0
converges to zero.
Lemma 2.6 see 8. Assume that X has a weakly continuous duality map J
ϕ
with gauge ϕ. Then,
A is demiclosed in the sense that A is closed in the product space X
w
× X,whereX is equipped with
the norm topology and X
w
with the weak topology. That is, if x
n
,y
n
∈ A, x
n
x, y
n
→ y, then
x, y ∈ A.
Lemma 2.7 see 12. Let X be a Banach space and γ ≥ 2. Then,
i X is uniformly convex if and only if, for any positive number r, there is a strictly increasing
continuous function g
r
: 0, ∞ → 0, ∞,g
r
00, such that
tx 1 − ty
γ
≤ tx
γ
1 − ty
γ
− W
γ
tg
r
x − y, 2.7
where t ∈ 0, 1,x,y∈ B
r
: {u ∈ X : u≤r}, the closed ball of X centered at the origin
with radius r, and W
γ
tt
γ
1 − tt1 − t
γ
.
ii X is γ-uniformly convex if and only if there holds the inequality
tx 1 − ty
γ
≤ tx
γ
1 − ty
γ
− c
γ
W
γ
tx − y
γ
,t∈ 0, 1,x,y∈ X, 2.8
where c
γ
> 0 is a constant.
Lemma 2.8 see 4. Let C be a closed convex subset of a uniformly convex Banach space with
aFr
´
echet differentiable norm, and let T
n
be a sequence of nonexpansive self mapping of C with a
nonempty common fixed point set F. If x
1
∈ C and x
n1
T
n
x
n
for n ≥ 1, then lim
n →∞
x
n
,Jf
1
−
f
2
exists for all f
1
,f
2
∈ F. In particular, q
1
− q
2
,Jf
1
− f
2
0, where f
1
,f
2
∈ F and q
1
,q
2
are
weak limit points of {x
n
}.
Lemma 2.9 the demiclosedness principle of nonexpansive mappings 13. Let T be a
nonexpansive selfmapping of a closed convex subset of E of a uniformly convex Banach space. Suppose
that T has a fixed point. Then I − T is demiclosed. This means that
{x
n
}⊂E, x
n
x, I − Tx
n
→ y ⇒ I − Tx y. 2.9
In 2005, Kim and Xu 7 , proved the following theorem.
Fixed Point Theory and Applications 5
Theorem A. Let C be a closed convex subset of a uniformly smooth Banach space X, and let T : C →
C be a nonexpansive mapping such that FixT
/
∅. Given a point u ∈ C and given sequences {α
n
}
∞
n0
and {β
n
}
∞
n0
in 0, 1, the following conditions are satisfied.
i α
n
→ 0,β
n
→ 0,
ii
∞
n0
α
n
∞,
∞
n0
β
n
∞,
iii
∞
n0
|α
n1
− α
n
| < ∞,
∞
n0
|β
n1
− β
n
| < ∞.
Define a sequence {x
n
}
∞
n0
in C by
x
0
∈ C arbitrarily,
y
n
α
n
x
n
1 − α
n
Tx
n
,n≥ 0,
x
n1
β
n
u
1 − β
n
y
n
,n≥ 0.
2.10
Then {x
n
}
∞
n0
is strongly converges to a fixed point of T.
Recently, the study of fixed points by doubly Mann iteration process began by Moore
see 14.In15, 16, we introduced the concept of Mann-type doubly sequence iteration
with errors, then we obtained some fixed point theorems for some different classes of
mappings. In this paper, we will continue our study in the doubly sequence setting. We
propose two modifications of the doubly Mann iteration process with errors in uniformly
smooth Banach spaces: one for nonexpansive mappings and the other for the resolvent of
accretive operators. The two modified doubly Mann iterations are proved to have strong
convergence. Also, we append this paper by obtaining weak convergence theorems for
Mann’s doubly sequence iteration with errors in a uniformly convex Banach space by a
Fr
´
echet differentiable norm. Our results in this paper extend, generalize, and improve a lot of
known results see, e.g., 4, 7, 8, 17. Our generalizations and improvements are in the use
of doubly sequence settings as well as by adding the error part in the iteration processes.
3. A Fixed Point of Nonexpansive Mappings
In this section, we propose a modification of doubly Mann’s iteration method with errors to
have strong convergence. Modified doubly Mann’s iteration process is a convex combination
of a fixed point in C, and doubly Mann’s iteration process with errors can be defined as
x
0,0
x ∈ C arbitrarily,
y
k,n
α
n
x
k,n
1 − α
n
Tx
k,n
α
n
w
k,n
,k,n≥ 0,
x
k,n1
β
n
u
1 − β
n
y
k,n
β
n
v
k,n
,k,n≥ 0.
3.1
The advantage of this modification is that not only strong convergence is guaranteed, but also
computations of iteration processes are not substantially increased.
Now, we will generalize and extend Theorem A by using scheme 3.1.
Theorem 3.1. Let C be a closed convex subset of a uniformly smooth Banach space X and let T :
C → C be a nonexpansive mapping such that FixT
/
∅. Given a point u ∈ C and given sequences
{α
n
}
∞
n0
and {β
n
}
∞
n0
in 0, 1, the following conditions are satisfied.
6 Fixed Point Theory and Applications
i α
n
→ 0,β
n
→ 0,
ii
∞
n0
α
n
∞,
∞
n0
β
n
∞.
Define a sequence {x
k,n
}
∞
k,n0
in C by 3.1. Then, {x
k,n
}
∞
k,n0
converges strongly to a fixed point of T.
Proof. First, we observe that {x
k,n
}
∞
k,n0
is bounded. Indeed, if we take a fixed point p of T
noting that
y
k,n
− p
α
n
x
k,n
1 − α
n
Tx
k,n
α
n
w
k,n
− p
≤ α
n
x
k,n
− p
1 − α
n
Tx
k,n
− p
α
n
w
k,n
x
k,n
− p
α
n
w
k,n
,
3.2
we obtain
x
k,n1
− p
β
n
u
1 − β
n
y
k,n
β
n
v
k,n
− p
≤ β
n
u − p
1 − β
n
y
k,n
− p
β
n
v
k,n
≤ β
n
u − p
1 − β
n
x
k,n
− p
α
n
w
k,n
β
n
v
k,n
≤ max
x
k,n
− p
, u − p
β
n
v
k,n
1 − β
n
α
n
w
k,n
.
3.3
Now, an induction yields
x
k,n
− p
≤ max
x
0,0
− p
, u − p,
v
0,0
k, n ≥ 0. 3.4
Hence, {x
k,n
} is bounded, so is {y
k,n
}. As a result, we obtain by condition i
x
k,n1
− y
k,n
β
n
u − β
n
y
k,n
β
n
v
k,n
≤ β
n
u − y
k,n
β
n
v
k,n
−→ 0.
3.5
We next show that
x
k,n
− Tx
k,n
−→ 0. 3.6
It suffices to show that
x
k,n1
− x
k,n
−→ 0. 3.7
Indeed, if 3.7 holds, in view of 3.5,weobtain
x
k,n
− Tx
k,n
≤
x
k,n
− x
k,n1
x
k,n1
− y
k,n
y
k,n
− Tx
k,n
≤
x
k,n
− x
k,n1
x
k,n1
− y
k,n
α
n
x
k,n
− Tx
k,n
α
n
w
k,n
−→ 0.
3.8
Fixed Point Theory and Applications 7
Hence, 3.6 holds. In order to prove 3.7, we calculate
x
k,n1
− x
k,n
β
n
− β
n−1
u − Tx
n−1
1 − β
n
α
n
x
k,n
− x
k,n−1
α
n
− α
n−1
1 − β
n
−
β
n
− β
n−1
α
n−1
x
k,n−1
− Tx
k,n−1
1 − α
n
1 − β
n
Tx
k,n
− Tx
k,n−1
1 − β
n
α
n
w
k,n
β
n
v
k,n
−
1 − β
n
α
n−1
w
k,n−1
− β
n−1
v
k,n−1
.
3.9
It follows that
x
k,n1
− x
k,n
≤
1 − α
n
1 − β
n
Tx
k,n
− Tx
k,n−1
1 − β
n
α
n
x
k,n
− x
k,n−1
α
n
− α
n−1
1 − β
n
−
β
n
− β
n−1
α
n−1
x
k,n−1
− Tx
k,n−1
β
n
− β
n−1
u − Tx
k,n−1
1 − β
n
α
n
w
k,n
β
n
v
k,n
−
1 − β
n
α
n−1
w
k,n−1
− β
n−1
v
k,n−1
.
3.10
Hence, by assumptions i-ii,weobtainx
k,n1
− x
k,n
→0.
Next, we claim that
lim
k,n→∞
sup
u − q, J
x
k,n
− q
≤ 0, 3.11
where q Qus − lim
t → 0
z
t
with z
t
being the fixed point of the contraction z → tu 1 −
tTz. In order to prove 3.11, we need some more information on q, which is obtained from
that of z
t
cf. 18. Indeed, z
t
solves the fixed point equation
z
t
tu 1 − tTz
t
tv. 3.12
Thus we have
z
t
− x
k,n
1 − t
Tz
t
− x
k,n
t
u − x
k,n
tv. 3.13
We apply Lemma 2.3 to get
z
t
− x
k,n
2
≤ 1 − t
2
Tz
t
− x
k,n
2
2t
u v − x
k,n
,J
z
t
− x
k,n
≤
1 − 2t t
2
z
t
− x
k,n
a
n
t2t
u v − z
t
,J
z
t
− x
k,n
2t
z
t
− x
k,n
2
,
3.14
a
n
t
2
z
t
− x
k,n
x
k,n
− Tx
k,n
x
k,n
− Tx
k,n
−→ 0asn −→ ∞ . 3.15
It follows that
z
t
− u, J
z
t
− x
k,n
≤
t
2
z
t
− x
k,n
2
1
2t
a
n
t. 3.16
8 Fixed Point Theory and Applications
Letting n →∞in 3.16 and noting 3.15 yield
lim
n →∞
sup
z
t
− u, J
z
t
− x
k,n
≤
t
2
M, 3.17
where M>0 is a constant such that M ≥z
t
− x
k,n
2
for all t ∈ 0, 1 and n ≥ 1. Since the set
{z
t
− x
k,n
} is bounded, the duality map J is norm-to-norm uniformly continuous on bounded
sets of X Lemma 2.2,andz
t
strongly converges to q. By letting t → 0in3.17,thus3.11
is therefore proved. Finally, we show that x
k,n
→ q strongly and this concludes the proof.
Indeed, using Lemma 2.3 again, we obtain
x
k,n1
− q
2
1 − β
n
y
k,n
β
n
u β
n
v
k,n
− q
2
1 − β
n
y
k,n
− q
β
n
u − qβ
n
v
k,n
2
≤
1 − β
n
2
y
k,n
− q
2
2β
n
u v
k,n
− q, J
x
k,n1
− q
≤
1 − β
n
2
x
k,n
− q
α
n
w
k,n
2
2β
n
u v
n
− q, J
x
k,n1
− q
.
3.18
Now we apply Lemma 2.5,andusing3.11 we obtain that x
k,n
− q→0.
We support our results by giving the following examples.
Example 3.2. Let T : 0, 1 × 0, 1 → 0, 1 × 0, 1 be given by Tx x. Then, the modified
doubly Mann’s iteration process with errors converges to the fixed point x
∗
0, 0,andboth
Picard and Mann iteration processes converge to the same point too.
Proof. I Doubly Picards iteration converges.
For every point in 0, 1 × 0, 1 is a fi xed point of T. Let b
0,0
be a point in 0, 1 × 0, 1,
then
b
k1,k1
Tb
k,k
T
n
b
0,0
b
0,0
. 3.19
Hence,
lim
k →∞
b
k,k
b
0,0
. 3.20
Let x, y − a, b|x − a|, |y − b|, for all x, y, a, b ∈ 0, 1 × 0, 1. Take p
0,0
0, 0 and
p
k,k
1/k, 1/k. Thus
δ
k,k
p
k1,k1
− Tp
k,k
1
kk 1
,
1
kk 1
−→ 0, 0. 3.21
II Doubly Mann’s iteration converges.
Let e
0,0
be a point in 0, 1 × 0, 1, then
e
k1,k1
1 − α
k
e
k,k
α
k
e
k,k
e
k,k
··· e
0,0
. 3.22
Fixed Point Theory and Applications 9
Since doubly Mann’s iteration is defined by
e
k1,k1
1 − α
k
e
k,k
α
k
Te
k,k
. 3.23
Take u
0,0
e
0,0
,u
k,k
1/k 1, 1/k 1 to obtain
ε
k,k
u
k1,k1
−
1 − α
k
u
k,k
α
k
Tu
k,k
1
k 1k 2
,
1
k 1k 2
−→ 0, 0.
3.24
III Modified doubly Mann’s iteration process with errors converges because the
sequence e
k,k1
→ 0, 0 as we can see and by using 3.1,weobtain
y
k,k
α
k
e
k,k
1 − α
k
e
k,k
α
k
w
k,k
e
k,k
α
k
w
k,k
.
3.25
In 3.1, we suppose that u e
k,k
,
e
k,k1
β
k
u
1 − β
k
e
k,k
α
k
w
k,k
β
k
ν
k,k
e
k,k
1 − β
k
α
k
w
k,k
β
k
ν
k,k
,
e
k,k1
− e
k,k
1 − β
k
α
k
w
k,k
β
k
ν
k,k
.
3.26
Let k →∞and using Theorem 3.1 T is nonexpansive, we obtain e
k,k1
− e
k,k
0, 0.
Example 3.3. Let T : 0, ∞ × 0, ∞ → 0, ∞ × 0, ∞ be given by Tx x/4. Then the doubly
Mann’s iteration converges to the fixed point of x
∗
0, 0 but modified doubly Mann’s
iteration process with errors does not converge.
Proof. I Doubly Mann’s iteration converges because the sequence e
k,k
→ 0, 0 as we can
see,
e
k1,k1
1 − α
k
e
k,k
α
k
e
k,k
4
1 −
3α
k
4
e
k,k
n
m1
1 −
3α
m
4
e
0,0
≤ exp
−
3
4
n
k1
α
k
−→ 0, 0.
3.27
The last inequality is true because 1 − x ≤ exp−x, for all x ≥ 0and
n
k1
α
k
∞.
10 Fixed Point Theory and Applications
II The origin is the unique fixed point of T.
III Note that, modified doubly Mann’s iteration process with errors does not converge
to the fixed point of T, because the sequence e
k,k1
0, 0 as we can see and by
using 3.1,weobtain
y
k,k
α
k
e
k,k
1 − α
k
e
k,k
4
α
k
w
k,k
1 3α
k
4
e
k,k
α
k
w
k,k
. 3.28
Putting u e
k,k
,
e
k,k1
β
k
e
k,k
1 − β
k
1 3α
k
4
e
k,k
α
k
w
k,k
β
k
ν
k,k
. 3.29
Letting k →∞, we deduce that e
k,k1
0, 0.
4. Convergence to a Zero of Accretive Operator
In this section, we prove a convergence theorem for m-accretive operator in Banach spaces.
Let X be a real Banach space. Recall that, the possibly multivalued operator A with domain
DA and range RA in X is accretive if, for each x
i
∈ DA and y
i
∈ Ax
i
i 1, 2, there
exists a j ∈ Jx
2
− x
1
such that
y
2
− y
1
,j
≥ 0. 4.1
An accretive operator A is m-accretive if RIrAX for each r>0. Throughout this section,
we always assume that A is m-accretive and has a zero. The set of zeros of A is denoted by F.
Hence,
F {z ∈ DA :0∈ Az} A
−1
0. 4.2
For each r>0, we denote by J
r
the resolvent of A, that is, J
r
I rA
−1
. Note that if A
is m-accretive, then J
r
: X → X is nonexpansive and FixJ
r
F for all r>0. We need the
resolvent identity see 19, 20 for more information.
Lemma 4.1 7the resolvent identity. For λ>0, μ>0 and x ∈ X,
J
λ
x J
μ
μ
λ
x
1 −
μ
λ
J
λ
x
. 4.3
Theorem 4.2. Assume that X is a uniformly smooth Banach space, and A is an m-accretive operator
in X such that A
−1
0
/
∅. Let {x
k,n
} be defined by
x
0,0
x ∈ X,
y
k,n
J
r
n
x
k,n
,
x
k,n1
α
n
u
1 − α
n
y
k,n
α
n
w
k,n
.
4.4
Fixed Point Theory and Applications 11
Suppose {α
n
} and {r
n
} satisfy the conditions,
i lim
n →∞
α
n
0 and
∞
n0
α
n
∞,
ii
∞
n0
|α
n1
− α
n
| < ∞,
iii r
n
≥ ε for some ε>0 and for all n ≥ 1. Also assume that
∞
n1
1 −
r
n−1
r
n
< ∞. 4.5
Then, {x
k,n
} converges strongly to a zero of A.
Proof. First of all we show that {x
k,n
} is bounded. Take p ∈ F A
−1
0. It follows that
x
k,n1
− p
α
n
u
1 − α
n
J
r
n
x
k,n
α
n
w
k,n
− p
≤ α
n
u − p
1 − α
n
x
k,n
− p
α
n
w
k,n
.
4.6
By induction, we get that
x
k,n
− p
≤ max
x
0,0
− p
, u − p,
w
0,0
k, n ≥ 0. 4.7
This implies that {x
k,n
} is bounded. Then, it follows that
x
k,n1
− J
r
n
x
k,n
−→ 0. 4.8
A simple calculation shows that
x
k,n1
− x
k,n
α
n
− α
n−1
u − y
k,n−1
1 − α
n
y
k,n
− y
k,n−1
α
n
w
k,n
− α
n−1
w
k,n−1
.
4.9
The resolvent identity 4.3 implies that
y
k,n
J
r
n−1
r
n−1
r
n
x
k,n
1 −
r
n−1
r
n
J
r
n
x
k,n
, 4.10
12 Fixed Point Theory and Applications
which in turn implies that
y
k,n
− y
k,n−1
J
r
n−1
r
n−1
r
n
x
k,n
1 −
r
n−1
r
n
J
r
n
x
k,n
− J
r
n−1
x
k,n−1
,
J
r
n−1
r
n−1
r
n
x
k,n
− x
k,n−1
1 −
r
n−1
r
n
J
r
n
x
k,n
J
r
n−1
r
n−1
r
n
x
k,n
− x
k,n
x
k,n
− x
k,n−1
1 −
r
n−1
r
n
J
r
n
x
k,n
J
r
n−1
r
n−1
r
n
− 1
x
k,n
x
k,n
− x
k,n−1
1 −
r
n−1
r
n
J
r
n
x
k,n
1 −
r
n−1
r
n
J
r
n−1
− x
k,n
J
r
n−1
x
k,n
− x
k,n−1
≤
1 −
r
n−1
r
n
J
r
n
x
k,n
− J
r
n−1
x
k,n
J
r
n−1
x
k,n
− J
r
n−1
x
k,n−1
≤
1 −
r
n−1
r
n
J
r
n
x
k,n
− J
r
n−1
x
k,n
x
k,n
− x
k,n−1
.
4.11
Combining 4.9 and 4.11,weobtain
x
k,n1
− x
k,n
≤
1 − α
n
x
k,n
− x
k,n−1
M
α
n
− α
n−1
1 −
r
n−1
r
n
α
n
w
k,n
α
n−1
w
k,n−1
,
4.12
where M is a constant such that M ≥ max{u − y
k,n
, J
r
x
k,n
− x
k,n
} for all n ≥ 0andr>0.
By assumptions i–iii in the theorem, we have that lim
n →∞
α
n
0,
∞
n0
α
n
∞, and
|α
n
− α
n−1
| |1 − r
n−1
/r
n
| < ∞. Hence, Lemma 2.5 is applicable to 4.12, and we conclude
that x
k,n1
− x
k,n
→0.
Take a fi xed number r such that ε>r>0. Again from the resolvent identity 4.3,we
find
J
r
n
x
k,n
− J
r
x
k,n
J
r
r
r
n
x
k,n
1 −
r
r
n
J
r
n
x
k,n
− J
r
x
k,n
≤
1 −
r
r
n
x
k,n
− J
r
n
x
k,n
≤
x
k,n
− x
k,n1
x
k,n1
− J
r
n
x
k,n
−→ 0.
4.13
It follows that
x
k,n1
− J
r
x
k,n1
≤
x
k,n1
− J
r
n
x
k,n
J
r
n
x
k,n
− J
r
x
k,n
J
r
x
k,n
− J
r
x
k,n1
≤
x
k,n1
− J
r
n
x
k,n
J
r
n
x
k,n
− J
r
x
k,n
x
k,n
− x
k,n1
.
4.14
Fixed Point Theory and Applications 13
Hence,
x
k,n
− J
r
x
k,n
−→ 0. 4.15
Since in a uniformly smooth Banach space the sunny nonexpansive retract Q from X onto the
fixed point set FixJ
r
F A
−1
0 of J
r
is unique, it must be obtained from Reich’s theorem
Lemma 2.4. Namely, Qus − lim
t → 0
z
t
,u∈ X, where t ∈ 0, 1 and z
t
∈ X solve the fixed
point equation
z
t
− x
k,n
t
u − x
k,n
1 − t
J
r
x
t
− x
k,n
. 4.16
Applying Lemma 2.3,weget
z
t
− x
k,n1
2
1 − t
2
J
r
z
t
− x
k,n
2
2t
u − x
k,n
,J
z
t
− x
k,n
≤ 1 − t
2
z
t
− x
k,n
2
a
n
t2t
u − z
t
,J
z
t
− x
k,n
2t
z
t
− x
k,n
,
4.17
where a
n
t2z
t
− x
k,n
·J
r
x
k,n
− x
k,n
J
r
x
k,n
− x
k,n
2
→ 0by4.15. It follows that
z
t
− u, J
z
t
− x
k,n
≤
t
2
z
t
− x
k,n
2
1
2t
a
n
t. 4.18
Therefore, letting k, n →∞in 4.18,weget
lim
k,n→∞
sup
z
t
− u, J
z
t
− x
k,n
≤
t
2
M, 4.19
where M is a constant such that M ≥z
t
− x
k,n
2
for all t ∈ 0, 1 and n ≥ 1. Since z
t
→ Qu
strongly and the duality map J is norm-to-norm uniformly continuous on bounded sets of X,
it follows that by letting t → 0in4.19
lim
k,n→∞
sup
u − Qu,J
x
k,n
− Qu
≤ 0, 4.20
x
k,n1
− Qu
2
α
n
u − Qu
1 − α
n
J
r
n
x
k,n
− Qu
2
≤
1 − α
n
2
J
r
n
x
k,n
− Qu
2
2α
n
u − Qu,J
x
k,n1
− Qu
≤
1 − α
n
x
k,n
− Qu
2
2α
n
u − Qu,J
x
k,n1
− Qu
.
4.21
Now we apply Lemma 2.5 and using 4.20,weobtainthatx
k,n
− Qu→0.
14 Fixed Point Theory and Applications
5. Weakly Convergence Theorems
We next introduce the following iterative scheme. Given an initial x
0,0
∈ C, we define x
k,n
by
x
k,n1
α
n
x
k,n
1 − α
n
J
r
n
x
k,n
α
n
u
k,n
,k,n≥ 0. 5.1
Theorem 5.1. Let X be a uniformly convex Banach space with a Frech
´
et differentiable norm. Assume
that X has a weakly continuous duality map J
ϕ
with gauge ϕ. Assume also that
i α
n
→ 0,
ii r
n
→∞.
Then, the scheme 5.1 converges weakly to a point q in F.
Proof. First, we observe that for any p ∈ F, the sequence {x
k,n
− p} is nonincreasing.
Indeed, we have by nonexpansivity of J
r
n
,
x
k,n
− p
α
n
x
k,n
1 − α
n
J
r
n
x
k,n
α
n
u
k,n
− p
≤ α
n
x
k,n
− p
1 − α
n
J
r
n
x
k,n
− p
α
n
u
k,n
x
k,n
− p
α
n
u
k,n
.
5.2
In particular, {x
k,n
} is bounded, so is {J
r
n
x
k,n
}.LetW
w
x
k,n
be the set of weak limit point of
the sequence {x
k,n
}.
Note that we can rewrite the scheme 5.1 in the form
x
k,n1
T
n
x
k,n
,k,n≥ 0, 5.3
where T
n
is the nonexpansive mapping given by
T
n
x α
n
x
1 − α
n
J
r
n
x α
n
u, x ∈ C. 5.4
Then, we have FT
n
FJ
r
n
F for n ≥ 1. Hence, by Lemma 2.7,weget
q
1
− q
2
,J
f
1
− f
2
0,q
1
,q
2
∈ W
w
x
k,n
,f
1
,f
2
∈ F. 5.5
Therefore, {x
k,n
} will converge weakly to a point in F if we can show that W
w
x
k,n
⊂ F. To
show this, we take a point v in W
w
x
k,n
. Then we have a subsequence {x
k,n
i
} of {x
k,n
} such
that x
k,n
i
v.Notingthat
x
k,n1
− J
r
n
x
k,n
α
n
x
k,n
− α
n
J
r
n
x
k,n
α
n
u
k,n
≤ α
n
x
k,n
− J
r
n
x
k,n
α
n
u
k,n
−→ 0,
5.6
Fixed Point Theory and Applications 15
we obtain
A
r
n
i
−1
x
k,n
i
−1
⊂ AJ
r
n
i
−1
x
k,n
i
−1
,
A
r
n
i
−1
x
k,n
i
−1
−→ 0,J
r
n
i
−1
x
k,n
i
−1
v.
5.7
By Lemma 2.6, we conclude that 0 ∈ Av, that is, v ∈ F.
Theorem 5.2. Let X be a uniformly convex Banach space which either has a Frech
´
et differentiable
norm or satisfies Opial’s property. Assume for some >0,
i ≤ α
n
≤ 1 − for n ≥ 1,
ii r
n
≥ for n ≥ 1.
Then, the scheme 5.1 converges weakly to a point q in F.
Proof. We have shown that lim
k,n→∞
x
k,n
− p exists for all p ∈ F. Applying Lemma 2.7i,we
have a strictly increasing continuous function g : 0, ∞ → 0, ∞,g00, such that
x
k,n1
− p
2
α
n
x
k,n
1 − α
n
J
r
n
x
k,n
α
n
u
k,n
− p
2
α
n
x
k,n
− p
u
n
1 − α
n
J
r
n
x
k,n
− p
2
α
n
x
k,n
− p
2
α
n
u
k,n
2
1 − α
n
J
r
n
x
k,n
− p
2
− α
n
1 − α
n
g
x
k,n
− J
r
n
x
k,n
.
5.8
This implies that
α
n
1 − α
n
g
x
k,n
− J
r
n
x
k,n
≤
x
k,n
− p
2
−
x
k,n
− p
2
. 5.9
Since α
n
1 − α
n
≥
2
, we obtain by 5.9 that
k,n
g
x
k,n
− J
r
n
x
k,n
< ∞ ⇒ lim
k,n→∞
x
k,n
− J
r
n
x
k,n
0. 5.10
For any fixed λ ∈ 0, 1, by Lemma 4.1, we have
J
r
n
x
k,n
J
λ
λ
r
n
x
k,n
1 −
λ
r
n
J
r
n
x
k,n
. 5.11
We deduce that
J
r
n
x
k,n
− J
λ
x
k,n
≤
λ
r
n
x
k,n
1 −
λ
r
n
J
r
n
x
k,n
− x
k,n
1 −
λ
r
n
x
k,n
− J
r
n
x
k,n
≤
x
k,n
− J
r
n
x
k,n
−→ 0 n −→ ∞ .
5.12
16 Fixed Point Theory and Applications
Therefore we obtain by 5.9 that
x
k,n
− J
λ
x
k,n
−→ 0 n −→ ∞ ,λ∈ 0, 1. 5.13
Apply Lemma 2.9 to find out that W
w
x
k,n
⊂ FJ
λ
F. It remains to show that W
w
x
k,n
is
a singleton set. Towards this end, we take p, q ∈ W
w
x
k,n
and distinguish the two cases.
In case X has a Frech
´
et differentiable norm, we apply Lemma 2.8 to get
p − q, Jp − q 0, 5.14
hence, p q. In case X satisfies Opial’s condition, we can find two subsequences {x
k,n
i
},
{x
k,m
j
} such that x
k,n
i
p, x
k,m
j
q.If p
/
q, Opial’s property creates the contradiction,
lim
k,n→∞
x
k,n
− p
lim
k,n→∞
x
k,n
i
− p
< lim
k,n→∞
x
k,n
i
− q
lim
k,n→∞
x
k,m
j
− q
< lim
k,n→∞
x
k,n
j
− p
lim
k,n→∞
x
k,n
− p
.
5.15
In either case, we have shown that W
w
x
k,n
consists of exact one point, which is clearly the
weak limit of {x
k,n
}.
Remark 5.3. The schemes 3.1, 4.4,and5.1 generalize and extend several iteration
processes from literature see 7, 8, 17, 21–25 and others.
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